Locally Analytic Complete Cohomology

• Locally analytic complete cohomology groups are defined by extracting locally analytic vectors from the completed cohomology of Shimura varieties under p-adic Lie group actions.
• The construction employs the Hodge–Tate period map and perfectoid geometry to establish comparison isomorphisms with flag variety cohomology.
• These groups bridge automorphic, Galois, and Lie theoretic data, enhancing insights in p-adic Hodge theory and p-adic local Langlands correspondences.

Locally analytic complete cohomology groups constitute a modern framework in arithmetic geometry and representation theory that synthesizes completed etale (or coherent) cohomology of Shimura varieties and related rigid analytic spaces, filtered through the analytic vectors of $p$-adic Lie groups acting on their cohomological invariants. This construction generalizes classical continuous and locally analytic cohomology theories, and connects automorphic, Galois, and Lie-theoretic data via comparison isomorphisms with flag variety cohomology, perfectoid geometry, and analytic representation theory.

1. Foundational Definitions and Comparison Isomorphisms

A locally analytic complete cohomology group is, in characteristic $p$ settings, defined by first taking the completed cohomology (in the sense of Emerton) on an inverse system of Shimura varieties or similar arithmetic spaces, and then extracting the locally analytic vectors for the $p$-adic group action. Formally, for a fixed tame level $K^p$ and a Shimura variety of Hodge type, one has: $$\left(\widetilde{H}^i(K^p,\mathbb{Q}_p)\,\widehat{\otimes}_{\mathbb{Q}_p}\,\mathbb{C}_p\right)^{\mathrm{la}}$$ where $\widetilde{H}^i(K^p,\mathbb{Q}_p)$ is the completed cohomology (via inverse limits over varying $p$-levels), and the superscript “la” denotes locally analytic vectors for the $p$-adic group action. The main result is a natural isomorphism (Aoki, 14 Aug 2025): $$\left(\widetilde{H}^i(K^p,\mathbb{Q}_p)\,\widehat{\otimes}_{\mathbb{Q}_p}\,\mathbb{C}_p\right)^{\mathrm{la}} \cong H^i\left(\mathscr{F}\ell_\mu,\,\mathcal{O}_{K^p}^{\mathrm{la}}\right)$$ with similar statements for compactly supported cohomology. Here, $\mathscr{F}\ell_\mu$ is the flag variety associated to the Hodge cocharacter $\mu$ from the Shimura datum, and $\mathcal{O}_{K^p}^{\mathrm{la}}$ is the sheaf of locally analytic functions obtained by pushforward via the Hodge–Tate period map from the perfectoid Shimura variety.

2. Perfectoid Geometry, Hodge–Tate Period Map, and Sheaf Construction

The passage to perfectoid spaces is essential for the comparison isomorphism. Fixing $K^p$ sufficiently small, the inverse limit of the tower of Shimura varieties yields a perfectoid space $\mathcal{S}h_{K^p}^{\tor}$. The geometric input is then the Hodge–Tate period map: $\pi_{\mathrm{HT}^{\tor}}: \mathcal{S}h_{K^p}^{\tor} \rightarrow \mathscr{F}\ell_\mu$ which encodes the $p$-adic Hodge structure at infinite level. The analytic sections of the pushforward sheaf (of completed functions) along this map assemble to $\mathcal{O}_{K^p}^{\mathrm{la}}$, the automorphic sheaf on the flag variety representing locally analytic functions coming from the perfectoid Shimura variety. The resulting cohomology groups of these sheaves on $\mathscr{F}\ell_\mu$ recover, via the isomorphism above, the locally analytic vectors in the completed cohomology.

3. Implications for p-adic Local Langlands and Automorphic Theory

This framework generalizes prior results by Pan for the modular curve and Qiu-Su for unitary Shimura curves (Aoki, 14 Aug 2025). In these cases, completed cohomology and its analytic vectors correspond closely to geometric or automorphic data, and can be computed as flag variety cohomology. The current setting extends these constructions to arbitrary Shimura varieties of Hodge type, revealing a bridge between $p$-adic automorphic representations, completed cohomology, and geometric realizations on flag varieties.

For example, in the setting of unitary Shimura curves and a fixed embedding $\tau$, one obtains

$$\left(\widetilde{H}^1(K^{\wp},E)\,\widehat{\otimes}_{\mathbb{Q}_p}\,\mathbb{C}_p\right)^{\tau-\mathrm{la}} \cong H^1\left(\mathscr{F}\ell_\mu,\,\mathcal{O}_{K^p,E}^{\tau-\mathrm{la}}\right)$$

after appropriate passage to $K^{\wp}$-invariants.

4. Functorial Properties, Jacquet–Langlands, and Independence of Towers

Recent developments reveal that analytic functorial constructions, notably Scholze’s $p$-adic Jacquet–Langlands functor, commute with the passage to locally analytic vectors (Dospinescu et al., 26 Nov 2024). In dual local Shimura varieties (e.g., Lubin–Tate and Drinfeld towers), the locally analytic vectors in the cohomology of different period sheaves at infinite level are independent of the $p$-adic group actions of the two towers. This establishes canonical equivalences of analytic cohomology, compatibility with central characters, and ultimately isomorphisms of de Rham cohomology as smooth representations of the product group $G \times G_b$.

5. Geometric Sen Operators and Vanishing Results

The geometric and arithmetic Sen operator, defined via the Kodaira–Spencer isomorphism and the Hodge–Tate period map, governs the infinitesimal Galois/automorphic action on completed cohomology (see (Camargo, 2022)). The Sen operator can be computed as a universal pullback of equivariant vector bundles on the flag variety, corresponding to derived quotients in the adjoint and parabolic Lie algebra representations. This identification enables rational vanishing results for completed cohomology above the geometric dimension: $$\widetilde{H}^i(K^p,\mathbb{Z}_p)[1/p] = 0 \text{ for } i > d$$ where $d$ is the complex/geometric dimension of the Shimura variety. These vanishing results align with Calegari–Emerton conjectures, further embedding the analytic cohomology theory within the broader landscape of $p$-adic Hodge theory and automorphic forms.

6. Cohomological Realization and Classicality

For two-dimensional regular $\sigma$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $\sigma$-analytic vectors of completed cohomology, one can realize classical automorphic forms within the analytic structure (Qiu et al., 15 May 2025). The methods establish a locality theorem: the Galois representation arising in locally $\sigma$-algebraic vectors is necessarily $\sigma$-de Rham, and the geometric realization via such vectors is admissible for coherent cohomology of Drinfeld curves. Special cases of Breuil’s locally analytic $\mathrm{Ext}^1$-conjecture for $\mathrm{GL}_2(L)$ are also proven in this setting.

7. Role in Arithmetic and Representation Theory

Locally analytic complete cohomology groups provide an arithmetic-geometric invariant that not only encodes automorphic and Galois-theoretic information, but also translates naturally into the field of analytic representations of $p$-adic Lie groups. The comparison isomorphisms to flag variety cohomology allow explicit calculation and geometric interpretation, while the results on independence, functoriality, and vanishing situate the theory at the intersection of modern $p$-adic geometry, automorphic representation theory, and $p$-adic local Langlands correspondences.

Table: Comparison Isomorphisms

Invariant (Shimura variety)	Corresponding Flag Variety Cohomology	Reference
$$\left(\widetilde{H}^i(K^p,\mathbb{Q}_p)\,\widehat{\otimes}_{\mathbb{Q}_p}\,\mathbb{C}_p\right)^{\mathrm{la}}$$	$$H^i\left(\mathscr{F}\ell_\mu,\,\mathcal{O}_{K^p}^{\mathrm{la}}\right)$$	(Aoki, 14 Aug 2025)
Compactly supported: $\widetilde{H}_c^i$	$$H^i\left(\mathscr{F}\ell_\mu,\,\mathcal{I}_{K^p}^{\mathrm{la}}\right)$$	(Aoki, 14 Aug 2025)

References and Context

For further foundational results on locally analytic cohomology, including duality and comparison to Lie algebra cohomology, see (Lechner, 2012, Tamme, 2014, Thomas, 2020), and (Fust, 2023). The geometric approach rests on the theory of perfectoid spaces, Sen operators, and analytic stacks, with deep implications for $p$-adic Hodge theory and automorphic representations.

In sum, locally analytic complete cohomology groups form a technically robust and geometrically rich invariant at the intersection of arithmetic geometry, representation theory, and $p$-adic analysis, generalizing classical constructions and enabling new correspondences central to modern number theory.